On the stability of global solutions to the three-dimensional Navier-Stokes equations

Hajer Bahouri; Jean-Yves Chemin; Isabelle Gallagher

doi:10.5802/jep.84

On the stability of global solutions to the three-dimensional Navier-Stokes equations
[Sur la stabilité de solutions globales aux équations de Navier-Stokes tridimensionnelles]

Hajer Bahouri ¹ ; Jean-Yves Chemin ² ; Isabelle Gallagher ³

¹ Laboratoire d’Analyse et de Mathématiques Appliquées UMR 8050, Université Paris-Est Créteil 61, avenue du Général de Gaulle, 94010 Créteil Cedex, France
² Laboratoire Jacques Louis Lions - UMR 7598, Sorbonne Université Boîte courrier 187, 4 place Jussieu, 75252 Paris Cedex 05, France
³ DMA, École normale supérieure, CNRS, PSL Research University 75005 Paris and UFR de mathématiques, Université Paris-Diderot, Sorbonne Paris-Cité 75013 Paris, France

Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 843-911.

Résumé
Abstract

On démontre un résultat de stabilité faible pour les équations de Navier-Stokes tridimensionnelles, incompressibles et homogènes. Plus précisément on étudie le problème suivant : si une suite de données initiales ${(u_{0, n})}_{n \in ℕ}$ , bornée dans un espace invariant d’échelle, converge faiblement vers une donnée $u_{0}$ qui engendre une solution globale régulière, est-ce que $u_{0, n}$ engendre une solution globale régulière ? Une réponse affirmative à cette question en général aurait pour conséquence la régularité globale pour toute donnée initiale, via les exemples $u_{0, n} = n ϕ_{0} (n \cdot)$ ou $u_{0, n} = ϕ_{0} (\cdot - x_{n})$ avec $| x_{n} | \to \infty$ . On introduit donc un nouveau concept de convergence faible (convergence faible remise à l’échelle) sous lequel on peut donner une réponse affirmative. La démonstration repose sur des décompositions en profils dans des espaces de régularité anisotrope, et leur propagation par les équations de Navier-Stokes.

We prove a weak stability result for the three-dimensional homogeneous incompressible Navier-Stokes system. More precisely, we investigate the following problem: if a sequence ${(u_{0, n})}_{n \in ℕ}$ of initial data, bounded in some scaling invariant space, converges weakly to an initial data $u_{0}$ which generates a global smooth solution, does $u_{0, n}$ generate a global smooth solution? A positive answer in general to this question would imply global regularity for any data, through the following examples $u_{0, n} = n ϕ_{0} (n \cdot)$ or $u_{0, n} = ϕ_{0} (\cdot - x_{n})$ with $| x_{n} | \to \infty$ . We therefore introduce a new concept of weak convergence (rescaled weak convergence) under which we are able to give a positive answer. The proof relies on profile decompositions in anisotropic spaces and their propagation by the Navier-Stokes equations.

Reçu le : 2018-02-13
Accepté le : 2018-10-13
Publié le : 2018-11-05

MR Zbl

DOI : 10.5802/jep.84

Classification : 35Q30, 42B37
Keywords: Navier-Stokes equations, anisotropy, Besov spaces, profile decomposition
Mot clés : Équations de Navier-Stokes, anisotropie, espaces de Besov, décompositions en profils

Affiliations des auteurs :

Hajer Bahouri ¹ ; Jean-Yves Chemin ² ; Isabelle Gallagher ³

¹ Laboratoire d’Analyse et de Mathématiques Appliquées UMR 8050, Université Paris-Est Créteil 61, avenue du Général de Gaulle, 94010 Créteil Cedex, France
² Laboratoire Jacques Louis Lions - UMR 7598, Sorbonne Université Boîte courrier 187, 4 place Jussieu, 75252 Paris Cedex 05, France
³ DMA, École normale supérieure, CNRS, PSL Research University 75005 Paris and UFR de mathématiques, Université Paris-Diderot, Sorbonne Paris-Cité 75013 Paris, France

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {On the stability of global solutions to the three-dimensional {Navier-Stokes} equations},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Hajer Bahouri; Jean-Yves Chemin; Isabelle Gallagher. On the stability of global solutions to the three-dimensional Navier-Stokes equations. Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 843-911. doi : 10.5802/jep.84. https://jep.centre-mersenne.org/articles/10.5802/jep.84/

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